In this paper, we derive a boundary and an internal observability inequality for stochastic hyperbolic equations with nonsmooth lower order terms. The required inequalities are obtained by global Carleman estimate for stochastic hyperbolic equations. By these inequalities, we study a state observation problem for stochastic hyperbolic equations. As a consequence, we also establish a unique cont...

We formulate a quasilinear parabolic equation describing the behavior of global-in-time solution to semilinear equation. study this in accordance with blow-up and quenching patterns original This is new theory partial differential equations presents several difficulties for mathematical analysis. Two approaches are examined: functional analysis viscosity solution.

Geometric analysis can be said to originate in the 19’th century work of Weierstrass, Riemann, Schwarz and others on minimal surfaces, a problem whose history can be traced at least as far back as the work of Meusnier and Lagrange in the 18’th century. The experiments performed by Plateau in the mid-19’th century, on soap films spanning wire contours, served as an important inspiration for this...

We study the Cauchy problem for 22 semilinear and quasilinear hyperbolic systems with a singular relaxation term. Special comparison and compactness properties are established by assuming the subcharacteristic condition. Therefore we can prove the convergence to equilibrium of the solutions of these problems as the singular perturbation parameter tends to zero. This research was strongly motiva...

We propose in this paper a new nonlinear mathematical model of an oscillating water column (OWC). The one-dimensional shallow equations the presence device are reformulated as transmission problem related to interaction between waves and fixed partially immersed structure. By imposing conservation total fluid-OWC energy non-damped scenario, we able derive condition that involves time-dependent ...

Balance laws arise from many areas of engineering practice specifically from the fluid mechanics. Many numerical methods for the solution of these balanced laws were developed in recent decades. The numerical methods are based on two views: solving hyperbolic PDE with a nonzero source term (the obvious description of the central and central-upwind schemes; (Kurganov & Levy, 2002; LeVeque, 2004)...

We propose a new regularization method for constructing a shock wave type solution with nonsmooth front (interaction of shock waves) for quasilinear equations in the onedimensional case.